How to play Reverse Hex

We present new results on how to play Reverse Hex, also known as Rex, or Misère Hex, on n × n boards. We give new proofs–and strengthened versions–of Lagarias and Sleator’s theorem (for n × n boards, each player can prolong the game until the board is full, so the first/second player can always win if n is even/odd) and Evans’s theorem (for even n , the acute corner is a winning opening move for the first player). Also, for even n ≥ 4 , we find another first-player winning opening (adjacent to the acute corner, on the first player’s side), and for odd n ≥ 3 , and for each first-player opening, we find second-player winning replies. Finally, in response to comments by Martin Gardner, for each n ≤ 5 , we give a simple winning strategy for the n × n board.